The recent fiscal-cliff showdown has left Republicans smarting, putting much of their legislative gamesmanship over the spending and taxes standoff at odds with the public's desire for compromise. A newly-released Pew poll suggests that the messy negotiations benefited the White House: Just 19 percent gave GOP leaders good marks on their handling of the negotiations, while 48 percent did the same for President Obama. No one particularly cared for the final deal, with only 38 percent approving and 41 percent disapproving.

And while House Speaker John Boehner and Senate Minority Leader Mitch McConnell live to fight another day, utilizing their next best piece of legislative leverage -- the debt ceiling -- may also come with a political price tag.

Former House Speaker Newt Gingrich called using the debt ceiling as leverage a "dead loser," and said that House Republicans need to find "a totally new strategy."

But Republican strategists believe that even another déjà vu showdown over sequestration and the debt ceiling won’t carry the same costs, with party leaders confident that in this next round, the public is more likely to be on their side than they were during the fiscal-cliff negotiations. They argue that Democrats got their tax hikes on the wealthy -- something Obama campaigned on, and something Republicans eventually caved on -- while no spending cuts were enacted.

“A debate driven by tax rates for the wealthy was a much easier debate for [Obama] than a debate about debt limits and deficits and spending,” says Kevin Madden, a GOP strategist and former Mitt Romney campaign adviser.

House Republican leadership has cited to its caucus a December poll commissioned by the GOP firm the Winston Group showing the public is largely against raising the debt ceiling, with 60 percent opposed and just 30 percent in favor.

“The Republicans don’t have to change their stripes to succeed. They need to leverage public opinion about spending and the deficits,” says veteran Republican strategist and former Boehner aide Terry Holt. “The American people are there -- they expect cuts. They anticipate some level of austerity-reform.”

The ways things play out in Washington will also have just as much to do with the political implications faced by individual lawmakers as opposed to those faced by the party as a whole.

“At the end of the day, the only poll that matters is the one in people’s districts. I’m focused on the people in my district," Rep. Tim Griffin, R-Ark., said. "National polls include people in Nancy Pelosi’s district, Henry Waxman’s district.... I don’t work for them, and I’m not real worried about the national polls.”

For lawmakers such as Griffin, a debt-ceiling debate is one in a series of opportunities to push a broad conversation about the national debt and deficit reduction. "If we communicate effectively, I think we will be able to demonstrate to the American people that the president’s talk about a balanced approach was nonsense," Griffin said.

Moving forward, a number of Republicans have signaled a strong stance on the debt ceiling. Sen. Ted Cruz of Texas has said that when it comes to the upcoming debt-ceiling fight, "I don’t think what Washington needs is more compromise, I think what Washington needs is more common sense and more principle." Sen. Pat Toomey of Pennsylvania has said that his party needs "to be willing to tolerate a temporary, partial government shutdown."

“The reason why conservative Republicans have been pretty firm in their resolve is they’ve seen the evolution of the issue,” Holt says. In the 1990s, for instance, not many Americans cared about debt as a broad topic, Holt says. Now, the nation’s debt is much higher, and it was a top campaign issue.

Even so, it appears that Republican leadership is providing some wiggle room on the debt ceiling and trying to incorporate the issue into a broader conversation about debt reduction, spending cuts, and reform -- which may play better politically. Boehner said in recent Wall Street Journalinterview that the debt ceiling is “one point of leverage” but it’s “not the ultimate leverage.”

“As a party, we have to defend our core fiscal-conservative principles and I think the party as a whole recognizes that no one event is going to result in the reformation for the Republican Party, but it’s part of a process,” Madden says. “But we’ll keep aligning ourselves as the stewards of good fiscal policy.”

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